More is Less: Inducing Sparsity via Overparameterization

TL;DR

This paper demonstrates that overparameterized neural networks trained with gradient descent tend to find sparse solutions, improving understanding of implicit bias and sample complexity in compressed sensing.

Contribution

It introduces a novel analysis showing gradient flow converges to minimal -norm solutions in sparse recovery, enhancing theoretical understanding of overparameterization.

Findings

Gradient flow converges to -minimal solutions when reconstructing sparse vectors.

Overparameterization improves sample complexity in compressed sensing.

Theoretical predictions match numerical experiments.

Abstract

In deep learning it is common to overparameterize neural networks, that is, to use more parameters than training samples. Quite surprisingly training the neural network via (stochastic) gradient descent leads to models that generalize very well, while classical statistics would suggest overfitting. In order to gain understanding of this implicit bias phenomenon we study the special case of sparse recovery (compressed sensing) which is of interest on its own. More precisely, in order to reconstruct a vector from underdetermined linear measurements, we introduce a corresponding overparameterized square loss functional, where the vector to be reconstructed is deeply factorized into several vectors. We show that, if there exists an exact solution, vanilla gradient flow for the overparameterized loss functional converges to a good approximation of the solution of minimal $\ell_1$-norm. The latter is well-known to promote sparse solutions. As a by-product, our results significantly improve the sample complexity for compressed sensing via gradient flow/descent on overparameterized models derived in previous works. The theory accurately predicts the recovery rate in numerical experiments. Our proof relies on analyzing a certain Bregman divergence of the flow. This bypasses the obstacles caused by non-convexity and should be of independent interest.